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\def\mon{\diagxyto/ >->/}
\def\monleft{\diagxyto/<-< /}



\def\qcomment#1#2{\phantom{#1}\red{$\ot$ #2}}

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\def\tocline#1#2{\par #1 \dotfill #2}
\def\tocline#1#2{\par \anchortargettext{.#2}{#2}{#1} \dotfill #2}
\def\myslide#1#2{\newpage{\bf #1}\par\label{#2}\addtolos{#1}}
  {\bf \anchortargettext{\the\count0}{.\the\count0}{\arabic{page}. #1}}


Sheaves over finite DAGs may be archetypal

(Or: ``Sheaves for non-categorists''. \red{Work in progress})


Eduardo Ochs - PURO/UFF





Presented at the XVI EBL


held at Petrópolis, RJ, Brazil, on 2011may13.

\red{These slides will probably be updated soon}

\red{to make them more self-contained.}


% (eedn4-51-bounded)

Index of the slides:
% To update the list of slides uncomment this line:
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")

\tocline {Let's mystify the audience with technical terms} {3}
\tocline {Let's mystify the audience a bit more} {4}
\tocline {Well-positioned subsets of $Z^2$ (and ZSets)} {5}
\tocline {Black pawn's moves (and ZDags)} {6}
\tocline {Partial orders} {7}
\tocline {Cycles are evil} {8}
\tocline {DAGs are good} {9}
\tocline {Our favorite topological space: ${\mathbb {V}}$} {10}
\tocline {Our favorite sheaves and presheaves} {11}
\tocline {Compatibility} {12}
\tocline {The evil presheaf} {13}
\tocline {Stack operations} {14}
\tocline {Stack operations (2)} {15}
\tocline {Covers and families} {16}
\tocline {Saturated families} {17}
\tocline {Adding unions} {18}
\tocline {Priming} {19}
\tocline {What next?} {20}


% --------------------
% «mention-modal»  (to ".mention-modal")
% (s "Let's mystify the audience with technical terms" "mention-modal")
\myslide {Let's mystify the audience with technical terms} {mention-modal}


{\bf Modal logic:}
\par S4 has the finite model property.
\par We have Gödel's translation: intuitionistic logic $\to$ S4
\par So: as $P ⊃ P$ is not a theorem of intutionistic logic
\par $\funto$ there is a finite model (with two worlds)
\par \lph{$\funto$} in which $P⊃P$ is not true.
\par These finite counter-models are good for developing
\par intuition about intuitionistic logic.
\par {\bf Category Theory:}
\par Let $\bbW$ be a finite poset.
\par ($\bbW$ is a system of possible worlds for S4,
\par viewed as a category).
\par Then $\Set^\bbW$ is a topos of presheaves.
\par The logic of toposes is intuitionistic,
\par and in $\Set^\bbW = \Set^{*\to*}$ we can falsify $P⊃P$.
\par Claim:
\par Toposes of the form $\Set^\bbW$ are good for developing
\par intuition about Topos Theory (and CT in general).

% --------------------
% «mention-sheaves»  (to ".mention-sheaves")
% (s "Let's mystify the audience a bit more" "mention-sheaves")
\myslide {Let's mystify the audience a bit more} {mention-sheaves}
\par Sheaves are very important in Topos Theory.
\par Category Theory is \red{hard} (too abstract).
\par Even \red{basic sheaf theory} is \red{too hard}.
\par Idea: Let's use toposes of the form $\Set^\bbW$
\par to learn about sheaves!
\par In ``Internal Diagrams in Category Theory'' (2010)
\par I ``defined'' (loosely) a way of thinking
\par diagrammatically, and a notion of how much
\par ``mental space'' each idea takes.
\par Specializations behave like projections,
\par Generalizations behave like liftings:

%L defarrows "<-- <- <-| <."

%D diagram spec-and-gen
%D 2Dx     100  +70
%D 2D  100 G    T
%D 2D
%D 2D  +50 P    S
%D 2D
%D (( G .tex= \arr{general\\theory}
%D    P .tex= \arr{particular\\case}
%D    T .tex= \arr{Toposes}
%D    S .tex= \arr{`$\Set^\bbW$'s}
%D    G P ->  sl_ .plabel= l \arr{specialization\\(projection)}
%D    G P <-- sl^ .plabel= r \arr{generalization\\(lifting)}
%D    T S ->  sl_
%D    T S <-- sl^
%D ))
%D enddiagram

\par Can we learn/define/understand sheaves
\par in toposes of the form $\Set^\bbW$
\par and then lift the theory to the general case?

% --------------------
% «zsets»  (to ".zsets")
% (s "Well-positioned subsets of $Z^2$ (and ZSets)" "zsets")

\myslide {Well-positioned subsets of $Z^2$ (and ZSets)} {zsets}
\par Def: a subset $D = \{(x_1,y_1),...,(x_n,y_n)\} \subset \Z^2$
\par is {\bf well-positioned} when $\inf_i x_i = 0$ and $\inf_i y_i = 0$.
\par Def: {\bf ZSet} is a finite well-positioned subset of $\Z^2$.
\par Examples: 
\par $Y = \{(0,2),(2,2),(1,1),(1,0)\}$
\par $K = \{(1,3),\,(0,2),(2,2),\,(1,1),\,(1,0)\}$
\par They will usually be named according to their shapes
\par (`K' is for `Kite').

% --------------------
% «black-pawns-moves»  (to ".black-pawns-moves")
% (s "Black pawn's moves (and ZDags)" "black-pawns-moves")
\myslide {Black pawn's moves (and ZDags)} {black-pawns-moves}


%D diagram Kite
%D 2Dx     100 +20 +20
%D 2D  100     K1
%D 2D         /  \
%D 2D  +20 K2      K3
%D 2D         \  /
%D 2D  +20     K4
%D 2D          |
%D 2D  +20     K5
%D 2D
%D (( K1 .tex= (1,3)
%D    K2 .tex= (0,2)
%D    K3 .tex= (2,2)
%D    K4 .tex= (1,1)
%D    K5 .tex= (1,0)
%D    K1 K2 -> K1 K3 ->
%D    K2 K4 -> K3 K4 ->
%D    K4 K5 ->
%D ))
%D enddiagram
% (find-pspage "~/IMAGES/black_pawns_moves.eps")
  % \includegraphics[scale=0.35]{../IMAGES/black_pawns_moves.eps}


Let $K = \{(1,3),\,(0,2),(2,2),\,(1,1),\,(1,0)\}$.

Then the set of {\bf black pawn's moves} on $K$, $\BPM{K}$,

is the set of 5 arrows at the right.

Let $\bbK = (K, \BPM{K})$  \qcomment{mmm}{this a DAG.}


Every ZSet $D$ induces a DAG

$\bbD = (D, \BPM{D})$   \qcomment{mmmmmi}{this a {\bf ZDag}.}

% --------------------
% «partial-orders»  (to ".partial-orders")
% (s "Partial orders" "partial-orders")
\myslide {Partial orders} {partial-orders}
\par We are interested in S4 and categories, so
\par we like relations that are reflexive and transitive.
\par It is clumsy to draw $(Y, \BPM{Y}^*)$ (at the right),
\par so we'd like to make $(Y, \BPM{Y})$ (at the left)
\par stand for $(Y, \BPM{Y}^*)$.

%D diagram Y
%D 2Dx     100 +20 +20
%D 2D  100 Y1      Y2 
%D 2D         \  /
%D 2D  +20     Y3     
%D 2D          |
%D 2D  +20     Y4     
%D 2D
%D (( Y1 .tex= (0,2)
%D    Y2 .tex= (2,2)
%D    Y3 .tex= (1,1)
%D    Y4 .tex= (1,0)
%D    Y1 Y3 -> Y2 Y3 -> Y3 Y4 ->
%D    # Y1 loop (ul,ur) 
%D    # Y2 loop (ul,ur) 
%D    # Y3 loop (ul,ur) 
%D    # Y4 loop (dl,dr) 
%D    # Y1 Y4 -> Y2 Y4 ->
%D ))
%D enddiagram
%D diagram Y*
%D 2Dx     100 +25 +25
%D 2D  100 Y1      Y2 
%D 2D        \   /
%D 2D  +20     Y3     
%D 2D          |
%D 2D  +20     Y4     
%D 2D
%D (( Y1 .tex= (0,2)
%D    Y2 .tex= (2,2)
%D    Y3 .tex= (1,1)
%D    Y4 .tex= (1,0)
%D    Y1 Y3 -> Y2 Y3 -> Y3 Y4 ->
%D    Y1 loop (ul,ur) 
%D    Y2 loop (ul,ur) 
%D    Y3 loop (ul,ur) 
%D    Y4 loop (dl,dr) 
%D    Y1 Y4 -> Y2 Y4 ->
%D ))
%D enddiagram
  \two/->`<--/<400>^{\rm (saturate)}_{\rm (?)}

\par Let's say that two relations, $R$ and $S$,
\par are {\bf equivalent} if $R^* = S^*$.
\par The class $[R] = \sst{S}{S^* = R^*}$ has a top element,
\par $R^*$, obtained by a kind of saturation process
\par (transitive-reflexive closure).

% --------------------
% «cycles-are-evil»  (to ".cycles-are-evil")
% (s "Cycles are evil" "cycles-are-evil")
\myslide {Cycles are evil} {cycles-are-evil}
\par Let $T = (\{1,2,3\}, \{1,2,3\}^2)$ be the complete graph on $\{1,2,3\}$.
\par Then $[T]$ has two different minimal elements:

%D diagram 123
%D 2Dx     100 +15 +15
%D 2D  100 1       2
%D 2D
%D 2D  +20     3
%D 2D
%D (( 1 2 -> 2 3 -> 3 1 ->
%D ))
%D enddiagram
%D diagram 321
%D 2Dx     100 +15 +15
%D 2D  100 1       2
%D 2D
%D 2D  +20     3
%D 2D
%D (( 1 2 <- 2 3 <- 3 1 <-
%D ))
%D enddiagram
%D diagram 123*
%D 2Dx     100 +15 +15
%D 2D  100 1       2
%D 2D
%D 2D  +20     3
%D 2D
%D (( 1 2 -> sl^ 1 2 <- sl_ 1 loop (ur,ul)
%D    2 3 -> sl^ 2 3 <- sl_ 2 loop (ur,ul)
%D    3 1 -> sl^ 3 1 <- sl_ 3 loop (dr,dl)
%D ))
%D enddiagram
%D diagram loops-are-evil
%D 2Dx     100   +70
%D 2D  100 123
%D 2D
%D 2D  +20       123*
%D 2D
%D 2D  +20 321
%D 2D
%D (( 123 .tex= \diag{123} BOX
%D    321 .tex= \diag{321} BOX
%D    123* .tex= \diag{123*} BOX
%D    123 123* -> sl^ 123 123* <-- sl_
%D    321 123* -> sl^ 321 123* <-- sl_
%D ))
%D enddiagram


\par If we want to represent partial orders by minimal graphs
\par we will need to avoid these...
\par ``Reflexive'' arrows, i.e., those of the form $\aa \to \aa$
\par are (sort of) irrelevant, so let's ignore them:
\par Def: $R^\refl$ is $R$ plus all reflexive arrows.
\par Def: $R^\irr$ is $R$ minus all reflexive arrows.
\par Def: $R$ is acyclic when \red{$R^\irr$} has no cycles.
     \qcomment{mmm}{not standard!}
\par Then in each class $[R]$ either all elements are acyclic
\par or all are cyclic.

% (find-dn4ex "edrx08.sty" "diagprep")
% (find-dn4ex "edrxdnt.tex" "defdiag")
% (find-dn5file "preamble.lua")
% (find-dn5file "README")
% (find-LATEXfile "2011ebl-slides.dnt")
% (find-LATEXfile "2010diags.dnt" "{dict-1}")
% (find-LATEXfile "2011ebl-slides.dnt" "{loops-are-evil}")

% --------------------
% «dags-are-good»  (to ".dags-are-good")
% (s "DAGs are good" "dags-are-good")
\myslide {DAGs are good} {dags-are-good}
\par ``Acyclic'' for us is ``acyclic modulo reflexive arrows''...
\par Consider the set of DAGs on a finite set of vertices $A$.
\par The equivalence relation $R \sim S \iff R^*=S^*$
\par partitions it into equivalent classes that are ``diamond-shaped'',
\par i.e., ``everything between a top and a bottom element'':
\par $[R] = \sst{R'}{R^\ess \subseteq R' \subseteq R^*}$.
\par To build $R^\ess$ from $R$ we drop all ``non-essential arrows''.
\par (This is the dual of the saturation $R \mto R^*$).
\par Moral: we can represent finite partial orders canonically
\par by their minimal DAGs (that only have ``essential arrows'').
\par ZDags are finite, acyclic, and minimal. 8-)

% --------------------
% «fav-top-space»  (to ".fav-top-space")
% (s "Our favorite topological space: $\bbV$" "fav-top-space")
\myslide {Our favorite topological space: $\bbV$} {fav-top-space}
\par Here it is:
\par as a DAG, $\bbV = (V, \BPM{V}) = (\{\aa,\bb,\cc\}, \{(\aa\to\cc),(\bb\to\cc)\})$
\par as a partial order, $\bbV = (V, \BPM{V}^*)$
\par as a top. space, $\bbV = (X, \Opens(X))$
     \qcomment{mmm}{note the renaming!}
\par \lph{mm} $= (X, \{\{\aa,\bb,\cc\}, \{\aa,\cc\}, \{\bb,\cc\}, \{\cc\}, \{\}\})$
\par \lph{mm} $= (X, \{X, U, V, W, \emp\})$
     \qcomment{mmmmmi}{names for the open sets}
\par \lph{mm} $= (X, \{\dagVee111, \dagVee101, \dagVee011, \dagVee001, \dagVee000\})$
     \qcomment{mmmmn}{positional notation!}
\par We can think of it as a quotient topology on $\R$...

%D diagram V-as-quot
%D 2Dx     100 +15 +15     +30 +15 +15      +30 +15 +15 
%D 2D  100 \aa     \bb     qaa     qbb      paa     pbb 
%D 2D                                                   
%D 2D  +15     \cc             qcc              pcc     
%D 2D                                                   
%D 2D  +15     X               QX               pX      
%D 2D                                                   
%D 2D  +15 U       V       QU      QV       pU      pV  
%D 2D                                                   
%D 2D  +15     W               QW               pW      
%D 2D                                                   
%D 2D  +15    \emp             Qemp             pemp    
%D (( \aa \cc -> \bb \cc ->
%D ))
%D (( qaa .tex= (-,3]
%D    qbb .tex= [2,+)
%D    qcc .tex= (2,3)
%D    qaa qcc -> qbb qcc ->
%D ))
%D (( paa .tex= \dagVee*{}
%D    pbb .tex= \dagVee*{}
%D    pcc .tex= \dagVee*{}
%D    paa pcc -> pbb pcc ->
%D ))
%D (( X U -> X V ->  U W -> V W ->  W \emp ->
%D ))
%D (( QX .tex= (-,+)
%D    QU .tex= (-,3)
%D    QV .tex= (2,+)
%D    QW .tex= (2,3)
%D    Qemp .tex= \emp
%D    QX QU -> QX QV ->  QU QW -> QV QW ->  QW Qemp ->
%D ))
%D (( pX .tex= \dagVee111
%D    pU .tex= \dagVee101
%D    pV .tex= \dagVee011
%D    pW .tex= \dagVee001
%D    pemp .tex= \dagVee000
%D    pX pU -> pX pV ->  pU pW -> pV pW ->  pW pemp ->
%D ))
%D enddiagram
\par I draw $X$ on top because it ``covers'' the other open sets,
\par and because $\dagVee111$ is $$ (``Top'') in the Heyting algebra
\par (but $$ is also the terminal... the HA must $\bbK^\op$).
\par Surprise: $(\Opens(X), \supseteq^\ess)$ is a ZDag!

% --------------------
% «fav-sh-and-presh»  (to ".fav-sh-and-presh")
% (s "Our favorite sheaves and presheaves" "fav-sh-and-presh")
\myslide {Our favorite sheaves and presheaves} {fav-sh-and-presh}
\par Let's write $\Opens(\R)$ for $(\Opens(\R), \subseteq)$  \qcomment{mmn}{a category ($\nearrow\nwarrow$)}
\par and $\Opens(\R)^\op$ for $(\Opens(\R), \supseteq)$.     \qcomment{mmmmm}{another \;\;\;\;($\swarrow\searrow$)}
\par Then $\calC^  \Set^{\Opens(\R)^\op}$ is a \red{sheaf}.
\par Bad news: it is too big to visualize.
\par We write $\bbV \equiv \dagVee***$ and $\bbK = \bbV' \equiv \dagKite*****$.
\par Let's define presheaves $C^, E  \Set^\bbK$.
\par A {\bf} presheaf in $\Set^\bbD$ is just a functor from $\bbD$ to $\Set$.
\par \red{Sheafness} is \red{separatedness} plus \red{collatedness}.
\par $C^$ will obey both, and $E$ will fail both.
%D diagram Cfo
%D 2Dx     100 +15 +15
%D 2D  100     \foX
%D 2D
%D 2D  +15 \foU    \foV
%D 2D
%D 2D  +15     \foW
%D 2D
%D 2D  +15     \foe
%D 2D
%D (( \foX \foU -> \foX \foV ->
%D    \foU \foW -> \foV \foW ->
%D    \foW \foe ->
%D ))
%D enddiagram
$ C^
  \;\; = \;\;
  \;\; = \;\;

% --------------------
% «compatibility»  (to ".compatibility")
% (s "Compatibility" "compatibility")
\myslide {Compatibility} {compatibility}
\par Let $U=(-,3)$ and $V=(2,)$ (temporarily).
\par Let $f_U  \calC^(U,\R)$ and $f_V  \calC^(V,\R)$, in:
$ \diagCfo{\calC^(\R,\R)}
\par We say that two ``locally defined functions'', $f_U$ and $f_V$,
\par are {\bf compatible} iff they ``coincide wherever they're both
\par defined'' (in the example: on $(2,3)$).
\par More precisely: $f_U$ and $f_V$ are compatible iff $f_U|_{UV} = f_V|_{UV}$.
% \par (To generalize, use the morphisms: $^U_{U∧V}(f_U) = ^V_{U∧V}(f_V)$).
\par Sheafness means that every {\bf compatible family} $\{f_U, \ldots, f_V\}$
\par has exactly one glueing to an $f_{U\ldotsV}$
\par (collatedness guarantees existence of a glueing,
\par separatedness guarantees that there is at most one).

% --------------------
% «evil-presheaf»  (to ".evil-presheaf")
% (s "The evil presheaf" "evil-presheaf")
\myslide {The evil presheaf} {evil-presheaf}
% \msk
\par Here is the ``evil presheaf'', $E:\dagKite***** \to \Set$.
\par Note that everything here is given explicitly ---
\par restriction functions that are the images of black pawn's moves,
\par e.g., $^X_V:E(X) \to E(V)$, are {\sl drawn};
\par restriction functions like $^U_U$ are necessarily $= \id_{E(U)}$, and
\par restriction functions like $^X_W$ are obtained by composition.
\par Note (again!) that $E$ is a {\sl functor}.
% (find-dn4 "experimental.lua" "relphantom")
%L forths[".xtag="] = function ()
%L     local x, y = ds:pick(1).x, ds:pick(1).y
%L     local dx, tag = getwordasluaexpr(), getword()
%L     tex = "\\ph{"..tag.."}"
%L     ds[1] = storenode {x=x+dx, y=y, tex=tex, tag=tag}
%L   end
%D diagram evil-presheaf
%D 2Dx     100 +20 +20  +30 +20 +20
%D 2D  100     A0           B0     
%D 2D         /  \         /  \    
%D 2D        v    v       v    v   
%D 2D  +20 A1      A2   B1      B2 
%D 2D        \    /       \    /   
%D 2D         v  v         v  v    
%D 2D  +20     A3           B3     
%D 2D          |            |      
%D 2D          v            v      
%D 2D  +20     A4           B4     
%D 2D
  %D (( A0 .tex= E(X)
  %D    A1 .tex= E(U)
  %D    A2 .tex= E(V)
  %D    A3 .tex= E(W)
  %D    A4 .tex= E(\emp)
  %D    A0 A1 ->
  %D    A0 A2 ->
  %D    A1 A3 ->
  %D    A2 A3 ->
  %D    A3 A4 ->
  %D ))
%D (( B0 .tex= \{e_X,e'_X\}  place  .xtag= -6 e_X  .xtag= 12 e'_X
%D  # B1 .tex= \{e_U,e'_U\}  place  .xtag= -5 e_U  .xtag= 10 e'_U
%D    B1 .tex= \{e_U\}       place  .xtag=  0 e_U  .xtag=  0 e'_U
%D    B2 .tex= \{e_V,e'_V\}  place  .xtag= -5 e_V  .xtag= 11 e'_V
%D    B3 .tex= \{e_W\}
%D    B4 .tex= \{e_\emp\}
%D    e_X e_U ->  e'_X e'_U ->
%D    e_X e_V ->  e'_X e_V  ->
%D    e_U B3  ->  e'_U B3   ->
%D    e_V B3  ->  e'_V B3   ->
%D    B3 B4 ->
%D ))
%D enddiagram
$ E
  \;\; = \;\;
  \;\; = \;\;
  % (find-LATEXgrep "grep -nH -e color *.tex")
\par Then $\{e_U, e_V\}$ is a compatible family,
\par because $e_U|_{UV} := ^U_W(e_U) = e^W$ and $e_V|_{UV} := ^V_W(e_V) = e^W$,
\par but $\{e_U, e_V\}$ has two different glueings, $e_X$ and $e'_X$,
\par so separatedness doesn't hold in $E$...
\par Also, $\{e_U, e'_V\}$ is another compatible family,
\par and this one has {\sl no glueings}.
\par So collatedness also doesn't hold in $E$.

% --------------------
% «stack-ops»  (to ".stack-ops")
% (s "Stack operations" "stack-ops")
\myslide {Stack operations} {stack-ops}
\par The fastest way to formalize all this is by using {\bf stacks}.
\par (This is not the standard way at all! I learned it from
\par Harold Simmons's ``The point-free approach to sheafification''.)
\par This is $E$ as a stack:
\par $ÆE = E(X)E(U)E(V)E(W)E(\emp)$
\par We have an operation called ``extent'', $[e_U] = U$,
\par going from $ÆE$ to $=\{X,U,V,W,\emp\}$,
\par and a non-commutative `$$', heavily overloaded,
\par that behaves as {\sl restriction} when its left arg is in $ÆE$
\par and as {\sl intersection} when its left arg is in $$:
   UV     &:=& U∧V \\
            &=& W   \\
   Ue_V   &:=& U[e_V] \\
            &=& UV \\
            &=& W \\
   e_UV   &:=& e_U|_{([e_U]V)} \\
            &=& e_W \\
   e_Ue_V &:=& e_U|_{([e_U][e_V])} \\
            &=& e_W \\

% --------------------
% «stack-ops-2»  (to ".stack-ops-2")
% (s "Stack operations (2)" "stack-ops-2")
\myslide {Stack operations (2)} {stack-ops-2}
\par The `$$' also accepts sets as arguments,
\par with the usual conventions:
\par $\{a,b\}\{c,d\} = \{ac, ad, bc, bd\}$,
\par $a\{b,c\} = \{ab, ac\}$,
\par $\{a,b\}c = \{ac, bc\}$.
\par (Also: $[\{a,b\}] = \{[a],[b]\}$).

% --------------------
% «covers-and-families»  (to ".covers-and-families")
% (s "Covers and families" "covers-and-families")
\myslide {Covers and families} {covers-and-families}
\par Def: a {\bf cover} is a subset of $$. (Example: $\{U,V\}$)
\par Def: a {\bf family} is a subset of $ÆE$ ``where $[]$ is injective''.
\par Def: a {\bf compatible family} is a family ``where `$$' commutes''.
\par Example 1: $\{e_V,e'_V\}$ is not a family.
\par Example 2: $\{e_U,e_V\}$ is a compatible family.
\par Example 3: $\{e_X,e'_V\}$ is non-compatible family.
\par $\def\ph#1{\phantom{#1}}
\par Notation for covers: $\calU, \calV, \ldots$, where $\bigcup\calV = V$.
\par Notation for families: $e_\calU$, where $[e_\calU] = \calU$.
\par Def: a cover $\calU$ is (downward) {\bf saturated} when $\calU=\calU$.
\par Def: a family $e_\calU$ is (downward) {\bf saturated} when $e_\calU=e_\calU$.
\par Example 4: $\{U,V\} = \{U,V,W,\emp\}$.
\par Example 5: $\{e_U,e'_V\} = \{e_U,e'_V,e_W,e_\emp\}$.
\par Example 6: $e_X = \{e_X, e_U, e_V, e_W, e_\emp\}$.
\par Example 7: $e_X\{U,V\} = \{e_U, e_V, e_W, e_\emp\}$.

% --------------------
% «saturated-families»  (to ".saturated-families")
% (s "Saturated families" "saturated-families")
\myslide {Saturated families} {saturated-families}
\par Let's annotate saturated covers with a `$*$'.
\par So: $\calU$, $\calU'$, $\calU^*$, $\calU^{*\prime}$ are saturated families, 
\par possibly different, all ``covering $U$''.
\par Let's write the saturation operation, `$$', as `$()^*$',
\par and let's say that $\calU \approx \calV$ when $(\calU)^*=(\calV)^*$,
\par and write the equivalence classes as $[\calU]$.
\par On \red{finite DAGs} each equivalence class has both a top element
\par and a bottom element:
\par $[\calU] = \sst{\calU'}{(\calU)^ \subseteq \calU' \subseteq (\calU)^*}$.
\par The operation $(\calU)^$, that drops all ``non-essential open sets''
\par in a cover, is new...
\par and it also makes sense for families.
% \par Each \red{compatible} family can be saturated in a unique way
\par Examples:
\par $\{U,V,W\}^* = \{U,V,W,\emp\}$
\par $\{U,V,W\}^ = \{U,V\}$
\par $\{e_U,e_V,e_W\}^* = \{e_U,e_V,e_W,e_\emp\}$
\par $\{e_U,e_V,e_W\}^ = \{e_U,e_V\}$

% «.adding-unions»	(to "adding-unions")
% --------------------
% «adding-unions»  (to ".adding-unions")
% (s "Adding unions" "adding-unions")
\myslide {Adding unions} {adding-unions}
\par In a \red{sheaf} $F:\bbK \to \Set$ every compatible family
\par $f_\calU$ can be glued in a unique way to obtain a $f_U$,
\par and we can obtain $f_\calU$ back from $f_U$: $f_\calU = f_U\calU$.
\par To understand what is going on here we need
\par another notion of saturation...
\par The `$*$' saturation adds {\sl smaller opens sets} to a cover;
\par The `$**$' saturation also adds {\sl unions} to a cover.
%D diagram X**
%D 2Dx     100     +30        +50        +35     +25
%D 2D  100 X ----> {X} --*--> {X}*	 dX      dX*
%D 2D	     <----     <-o---  |^	 
%D 2D	                     *o||** 
%D 2D	                       v|	 
%D 2D  +30        {U,V} --*-> {U,V}*     dUV     dUV*
%D 2D	                <-o--       
%D ((   X   .tex= X
%D     {X}  .tex= \{X\}
%D     {X}* .tex= \{X,U,V,W,\emp\}
%D   {U,V}  .tex= \{U,V\}
%D   {U,V}* .tex= \{U,V,W,\emp\}
%D    X    {X}   -> sl^
%D    X    {X}   <- sl_
%D   {X}   {X}*  -> sl^ .plabel= a *
%D   {X}   {X}*  <- sl_ .plabel= b 
%D   {X}* {U,V}* --> sl_ .plabel= l *
%D   {X}* {U,V}* <-- sl^ .plabel= r **
%D  {U,V} {U,V}* -> sl^ .plabel= a *
%D  {U,V} {U,V}* <- sl_ .plabel= b 
%D ))
%D (( dX   .tex= \dagKite10000
%D    dX*  .tex= \dagKite11111
%D    dUV  .tex= \dagKite01100
%D    dUV* .tex= \dagKite01111
%D    dX  dX* -> sl^ .plabel= a *
%D    dX  dX* <- sl_ .plabel= b 
%D    dX* dUV* .> sl_ .plabel= l *
%D    dX* dUV* <. sl^ .plabel= r **
%D    dUV dUV* -> sl^ .plabel= a *
%D    dUV dUV* <- sl_ .plabel= b 
%D ))
%D enddiagram
\par $\diag{X**}$

% (find-34page 10 "Quotient topologies")
% (find-34     "" "quotient")
% (find-34page 12 "set of saturated covers")
% (find-34        "set of saturated covers")

% --------------------
% «priming»  (to ".priming")
% (s "Priming" "priming")
\myslide {Priming} {priming}

\input 2011ebl-defs.tex

\def\A{\bigGuill 123456}
  {\G 111111}  % 12
 {(\G 101111)} % 11
  {\G 011111}  % 10
  {\G 001111}  % 8
 {(\G 010111)} % 9
  {\G 001011}  % 7
 {(\G 000111)} % 5
 {(\G 001010)} % 6
  {\G 000011}  % 4
 {(\G 000010)} % 3
 {(\G 000001)} % 2
  {\G 000000}  % 1
% $$
% \bhbox{$\A$} \to \bhbox{$\B$}
% $$
%D diagram guill-priming
%D 2Dx     100   +70
%D 2D  100 A     B
%D 2D
%D 2D  +50 C     D
%D 2D
%D 2D  +20 E     F
%D 2D
%D (( A .tex= \A
%D    B .tex= \B
%D    C .tex= \dagGuill******
%D    D .tex= (\dagGuill******)'=\dagGuillPrime************
%D    A B `-> # .plabel= a \aa|->\{\aa\}^\coint
%D    C D `->
%D ))
%D enddiagram

\par To understand ``topological sheaves'' we take a DAG (e.g., $\bbV$)
\par and prime it \red{twice}; the operations `$**$' and `$*$' work on $\bbV''$.
\par For ``generic'' sheaves (``sheaves on a site'') we take any DAG $\bbD$
\par to play the role of $\bbV'$ and an operation `$*$' on $\bbD$
\par that obeys three rules (obeyed by `$**$', of course),
\par and from there on we treat what were ``open sets''
\par as ``truth-values'' (!!!), and the `$*$' as a modality (!!!!!).

% --------------------
% «what-next»  (to ".what-next")
% (s "What next?" "what next")
\myslide {What next?} {what next}
\par ...but that doesn't fit in 20 minutes! 8-(
\par Look for the complete version of these slides in my home page!
\par \red{Goodbye! 8-)} 

%:*|-*\vdash *
%:  ------   --------
%:  P|-P^*   P^*|-Q^*

% (find-34     "" "priming")
% (find-34page  5 "priming")

% (find-854     "" "prop-calc-in-a-hyp")
% (find-854page 88 "prop-calc-in-a-hyp")

% (find-34     "" "primingp")
% (find-34page 5 "priming")

% (find-34        "presheaves")
% (find-34page 14 "presheaves")
% (find-LATEX "2008graphs.tex")

% (find-LATEX "2011ebl-abs.tex")



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